This invention relates to magnetic field correction devices (i.e., passive shim arrangements) for correcting the magnetic field generated by a magnet for producing a uniform field in magnetic resonance imaging devices, etc.
Generally, the magnets for producing a uniform magnetic field are designed to generate a homogeneous field by itself. However, due to errors in workmanship and effects of adjacent iron bodies, it is difficult to attain the designed level of magnetic field uniformity. Thus, a magnetic field correction device, i.e., passive shim arrangement, is provided in the magnet for compensating for the errors in workmanship and effects of the iron bodies.
FIG. 8 is a perspective view of a conventional magnet for generating a uniform magnetic field. A magnet casing 1 accommodates a solenoid coil (not shown) for generating a homogeneous magnetic field within the magnet casing 1. Magnetic shim elements 2 are fixed upon the interior of the casing 1 by means of attachment jigs 3, and corrects the inhomogeneity of the field within the uniform field region 4. In what follows, the positions of the shim elements 2 are specified with respect to the coordinate axes 5.
FIGS. 9a and 9b are a perspective and an end view, respectively, of a single magnetic bar, showing the positional parameters thereof with respect to the coordinate axes. Generally, as described in detail below, two bars as shown in FIGS. 9a and 9b of distinct lengths are soldered to each other to form a magnetic shim element 2.
The bar 9 is axially aligned with the Z-axis. The magnetic bar 9 is magnetized by means of the external magnetic field 8 along the Z-axis. As a result, magnetic charges appear at the end surfaces 10 and 11, such that a correction field 12 is generated. The magnetic shim elements 2 each consisting of two magnetic bars 9 of appropriate dimensions thus obtained are disposed at appropriate positions upon the interior of the magnet casing 1, such that the non-homogeneity of the magnetic field within the magnetic field region 4 is corrected.
Next, the correction field 12 generated by a magnetic bar 9 is described in detail. The magnetic bar 9 is disposed parallel to the Z-axis at a circumferential attachment angle .phi. and an attachment radius a (see FIG. 9b), such that the end surfaces of the magnetic bar 9 form end angles .alpha.1 and .alpha.2 with respect to the origin and the Z-axis (see FIG. 9a). The point P at which the field is measured is positioned at a radius r which form an angle .PHI. with respect to the X-axis (see FIG. 9b) and angle .theta. with respect to the Z-axis (see FIG. 9b).
Then, the magnetic field Bz formed by the magnetic bar 9 at measurement point P is given by equation (1): ##EQU1## where K is a constant determined by the magnetic characteristic of the magnetic bar 9, A is the cross-sectional area of the magnetic bar 9, .epsilon..sub.m is the Neumann coefficient (.epsilon..sub.m =2 if m.noteq. 0 and .epsilon..sub.m =1 if m=0), and Pn.sup.m is the associated Legendre polynomial of degree n and order m.
Further, the following TABLE 1 shows the correspondence between the output components of the magnetic field in the spherical polar coordinates Bz.sup.nm (up to n=2) and the cartesian components expressed in the orthogonal coordinate system X, Y, Z.
TABLE 1 ______________________________________ (CORRESPONDENCE UP TO n = 2) COMPONENTS IN ORTHOGONAL n m COORDINATES ______________________________________ 1 0 Z 1 1 X or Y 2 0 Z.sup.2 2 1 ZX or ZY 2 2 X.sup.2 - Y.sup.2 or XY ______________________________________
As seen from equation (1), the number of the magnetic components generated by the magnetic bar 9 is infinite. However, since generally a&gt;r holds, the factor (r/a).sup.n becomes negligibly small for those terms for which the value of n is great. Thus, it suffices to consider the components for which the values of n and m are small: Bz.sup.11, Bz.sup.21, Bz.sup.22, Bz.sup.31, Bz.sup.32, Bz.sup.33, Bz.sup.41, Bz.sup.42, Bz.sup.43, Bz.sup.44, Bz.sup.51, Bz.sup.52, Bz.sup.53, Bz.sup.54. Thus, for example, the dimensions and position of the bar 9 which generate only the Bz.sup.11 component corresponding to the X-component are determined as described below in the paragraphs (i), (ii), and (iii), in accordance with the method described in Japanese Laid-Open Patent Application (Kokai) No. 3-39676.
(i) The magnetic bars are attached at eight circumferential attachment angles .phi. as given by equations (a) through (h): EQU .phi.=(.pi./2)((1/2)+(1/3)+(1/4)) (a) EQU .phi.=(.pi./2)((1/2)+(1/3)-(1/4)) (b) EQU .phi.=(.pi./2)((1/2)-(1/3)+(1/4)) (c) EQU .phi.=(.pi./2)((1/2)-(1/3)-(1/4)) (d) EQU .phi.=(.pi./2)(-(1/2)+(1/3)+(1/4)) (e) EQU .phi.=(.pi./2)(-(1/2)+(1/3)-(1/4)) (f) EQU .phi.=(.pi./2)(-(1/2)-(1/3)+(1/4)) (g) EQU .phi.=(.pi./2)(-(1/2)-(1/3)-(1/4)) (h)
such that the factor: cos m(.PHI.-.phi.) for m=2, 3, 4 vanishes and thus the components: Bz.sup.22, Bz.sup.32, Bz.sup.33, Bz.sup.42, Bz.sup.43, Bz.sup.44, Bz.sup.52, Bz.sup.53, and Bz.sup.54 vanish.
(ii) The end angles .alpha.1 and .alpha.2 of each magnetic bar 9 are selected such that they satisfy: .alpha.2=.pi.-.alpha.1. Then, the following equations (2a) and (2b) hold and the components Bz.sup.21 and Bz.sup.41 vanish: EQU [P.sub.3.sup.1 (cos.alpha.) sin.sup.4 .alpha.].sub..pi.-.alpha.1.sup..alpha.1 =0 (2a) EQU [P.sub.5.sup.1 (cos.alpha.) sin.sup.6 .alpha.].sub..pi.-.alpha.1.sup..alpha.1 =0 (2b)
Further, by selecting the end angles .alpha.1 and .alpha.2 of the two magnetic bars 9 at 33.88.degree. and 146.12.degree., respectively, and at 62.04.degree. and 117.96.degree., respectively, the components Bz.sup.51 of the two magnetic bars constituting a shim element are both eliminated.
(iii) Assume that the cross-sectional area of the magnetic bar 9 having end angles .alpha.1 and .alpha.2 at 33.88.degree. and 146.12.degree. respectively, and that of the magnetic bar 9 having end angles .alpha.1, .alpha.2 at 62.04.degree. and 117.96.degree. respectively, are represented by A1 and A2, respectively. Then the resultant component Bz.sup.31 of the two magnetic bars 9 is given by: EQU Bz.sup.31 cc A1{P4.sup.1 (cos 33.88.degree.)(sin 33.88.degree.).sup.5 }+A2{P4.sup.1 (cos 62.04.degree.)(sin 62.04.degree.).sup.5 }
Thus, by selecting the ratio A1/A2 at 7.16 as shown by the following equation (3), the component Bz.sup.31 vanishes. ##EQU2##
Thus, as shown in FIG. 10, a bar 13 having end angles 33.88.degree. and 146.12.degree. and a bar 14 having end angles 62.4.degree. and 117.96.degree. are soldered to each other by means of solder 15 to form a single shim element 2. The shim elements 2 are attached to the casing 3 so that the conditions (i), (ii), and (iii) are satisfied. Thus, the components other than the desired Bz.sup.11 is eliminated and only the negative X-component field is generated.
The above description relates to the shim elements for generating the Bz.sup.11 components. The shim elements for generating the Bz.sup.21 and the Bz.sup.22 components can be obtained in a similar manner. The ratio of the cross sectional areas A1/A2 of the two magnetic bars constituting a shim element and the two end angles .alpha.1 and .alpha.2 thereof for generating the Bz.sup.11, Bz.sup.21, and Bz .sup.22 components, respectively, are summarized in the following TABLE 2.
TABLE 2 ______________________________________ components ratio of area (A.sub.1 /A.sub.2) end angles (.alpha..sub.1 & .alpha..sub.2) ______________________________________ B.sub.z.sup.11 7.16 (33.88.degree., 146.12.degree.) (62.04.degree., 117.96.degree.) B.sub.z.sup.21 1.25 (40.09.degree., 106.57.degree.) (40.09.degree., 73.43.degree.) B.sub.z.sup.22 5.47 (36.69.degree., 140.31.degree.) (65.11.degree., 114.89.degree.) ______________________________________
The conventional magnetic field correction device as described above, however, has the following disadvantages:
(1) Since the each shim element consists of two magnetic bars 13 and 14, the cross sectional form of the shim element varies along its length. Thus, when a plurality of shim elements for generating the same magnetic component are bundled together, the effective volume occupied by the magnetic bars is reduced and hence the space is used ineffectively.
(2) The assembly of the shim elements requires the soldering of two magnetic bars at a precise relative position. This assembly step is time consuming.
(3) As shown in TABLE 2, the end angles .alpha.1 and .alpha.2 for respective field components are different from each other. Thus, it is difficult to combine the shim elements for generating different field components together.